brcgr - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > brcgr		Structured version Visualization version Unicode version

Theorem brcgr 25780

Description: The binary relation form of the congruence predicate. The statement

Cgr

should be read informally as "the

dimensional point

is as far from

is from

, or "the line segment

is congruent to the line segment

. This particular definition is encapsulated by Tarski's axioms later on. (Contributed by Scott Fenton, 3-Jun-2013.)

**Assertion**
Ref	Expression
brcgr	$/\$ $/\$ $/\$ Cgr

Distinct variable groups:

Proof of Theorem brcgr Dummy variables

are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		opex 4932	. . 3
2		opex 4932	. . 3
3		eleq1 2689	. . . . . 6
4	3	anbi1d 741	. . . . 5 $/\$ $/\$
5		fveq2 6191	. . . . . . . . . 10
6	5	fveq1d 6193	. . . . . . . . 9
7		fveq2 6191	. . . . . . . . . 10
8	7	fveq1d 6193	. . . . . . . . 9
9	6, 8	oveq12d 6668	. . . . . . . 8
10	9	oveq1d 6665	. . . . . . 7
11	10	sumeq2sdv 14435	. . . . . 6
12	11	eqeq1d 2624	. . . . 5
13	4, 12	anbi12d 747	. . . 4 $/\$ $/\$ $/\$ $/\$
14	13	rexbidv 3052	. . 3 $/\$ $/\$ $/\$ $/\$
15		eleq1 2689	. . . . . 6
16	15	anbi2d 740	. . . . 5 $/\$ $/\$
17		fveq2 6191	. . . . . . . . . 10
18	17	fveq1d 6193	. . . . . . . . 9
19		fveq2 6191	. . . . . . . . . 10
20	19	fveq1d 6193	. . . . . . . . 9
21	18, 20	oveq12d 6668	. . . . . . . 8
22	21	oveq1d 6665	. . . . . . 7
23	22	sumeq2sdv 14435	. . . . . 6
24	23	eqeq2d 2632	. . . . 5
25	16, 24	anbi12d 747	. . . 4 $/\$ $/\$ $/\$ $/\$
26	25	rexbidv 3052	. . 3 $/\$ $/\$ $/\$ $/\$
27		df-cgr 25773	. . 3 Cgr $/\$ $/\$
28	1, 2, 14, 26, 27	brab 4998	. 2 Cgr $/\$ $/\$
29		opelxp2 5151	. . . . . . . . . . 11
30	29	ad2antll 765	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$
31		simplrr 801	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$
32		eedimeq 25778	. . . . . . . . . 10 $/\$
33	30, 31, 32	syl2anc 693	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$
34	33	adantlr 751	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
35		oveq2 6658	. . . . . . . . . 10
36	35	sumeq1d 14431	. . . . . . . . 9
37	35	sumeq1d 14431	. . . . . . . . 9
38	36, 37	eqeq12d 2637	. . . . . . . 8
39	34, 38	syl 17	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
40		op1stg 7180	. . . . . . . . . . . . 13 $/\$
41	40	fveq1d 6193	. . . . . . . . . . . 12 $/\$
42		op2ndg 7181	. . . . . . . . . . . . 13 $/\$
43	42	fveq1d 6193	. . . . . . . . . . . 12 $/\$
44	41, 43	oveq12d 6668	. . . . . . . . . . 11 $/\$
45	44	oveq1d 6665	. . . . . . . . . 10 $/\$
46	45	sumeq2sdv 14435	. . . . . . . . 9 $/\$
47		op1stg 7180	. . . . . . . . . . . . 13 $/\$
48	47	fveq1d 6193	. . . . . . . . . . . 12 $/\$
49		op2ndg 7181	. . . . . . . . . . . . 13 $/\$
50	49	fveq1d 6193	. . . . . . . . . . . 12 $/\$
51	48, 50	oveq12d 6668	. . . . . . . . . . 11 $/\$
52	51	oveq1d 6665	. . . . . . . . . 10 $/\$
53	52	sumeq2sdv 14435	. . . . . . . . 9 $/\$
54	46, 53	eqeqan12d 2638	. . . . . . . 8 $/\$ $/\$ $/\$
55	54	ad2antrr 762	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
56	39, 55	bitrd 268	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
57	56	biimpd 219	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
58	57	expimpd 629	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
59	58	rexlimdva 3031	. . 3 $/\$ $/\$ $/\$ $/\$ $/\$
60		eleenn 25776	. . . . 5
61	60	ad2antll 765	. . . 4 $/\$ $/\$ $/\$
62		opelxpi 5148	. . . . . . . . 9 $/\$
63		opelxpi 5148	. . . . . . . . 9 $/\$
64	62, 63	anim12i 590	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$
65	64	adantr 481	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$
66	54	biimpar 502	. . . . . . 7 $/\$ $/\$ $/\$ $/\$
67	65, 66	jca 554	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
68		fveq2 6191	. . . . . . . . . . 11
69	68	sqxpeqd 5141	. . . . . . . . . 10
70	69	eleq2d 2687	. . . . . . . . 9
71	69	eleq2d 2687	. . . . . . . . 9
72	70, 71	anbi12d 747	. . . . . . . 8 $/\$ $/\$
73	72, 38	anbi12d 747	. . . . . . 7 $/\$ $/\$ $/\$ $/\$
74	73	rspcev 3309	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$
75	67, 74	sylan2 491	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
76	75	exp32 631	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$
77	61, 76	mpcom 38	. . 3 $/\$ $/\$ $/\$ $/\$ $/\$
78	59, 77	impbid 202	. 2 $/\$ $/\$ $/\$ $/\$ $/\$
79	28, 78	syl5bb 272	1 $/\$ $/\$ $/\$ Cgr

Colors of variables: wff setvar class

Syntax hints:

wi 4

wb 196 $/\$ wa 384

wceq 1483

wcel 1990

wrex 2913

cop 4183 class class class wbr 4653

cxp 5112

cfv 5888 (class class class)co 6650

c1st 7166

c2nd 7167

c1 9937

cmin 10266

cn 11020

c2 11070

cfz 12326

cexp 12860

csu 14416

cee 25768 Cgrccgr 25770

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013

This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-fal 1489 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-1st 7168 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-er 7742 df-map 7859 df-en 7956 df-dom 7957 df-sdom 7958 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-nn 11021 df-n0 11293 df-z 11378 df-uz 11688 df-fz 12327 df-seq 12802 df-sum 14417 df-ee 25771 df-cgr 25773

This theorem is referenced by: axcgrrflx 25794 axcgrtr 25795 axcgrid 25796 axsegcon 25807 ax5seglem3 25811 ax5seglem6 25814 ax5seg 25818 axlowdimlem17 25838 ecgrtg 25863

W3C validator